A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{ M }{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9} \omega$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$. At this instant the distance of the other mass from $O$ is

In an orbital motion, the angular momentum vector is

The potential energy of a particle of mass $m$ at a distance $r$ from a fixed point $O$ is given by $\mathrm{V}(\mathrm{r})=\mathrm{kr}^2 / 2$, where $\mathrm{k}$ is a positive constant of appropriate dimensions. This particle is moving in a circular orbit of radius $\mathrm{R}$ about the point $\mathrm{O}$. If $\mathrm{v}$ is the speed of the particle and $\mathrm{L}$ is the magnitude of its angular momentum about $\mathrm{O}$, which of the following statements is (are) true?

$(A)$ $v=\sqrt{\frac{k}{2 m}} R$

$(B)$ $v=\sqrt{\frac{k}{m}} R$

$(C)$ $\mathrm{L}=\sqrt{\mathrm{mk}} \mathrm{R}^2$

$(D)$ $\mathrm{L}=\sqrt{\frac{\mathrm{mk}}{2}} \mathrm{R}^2$

A particle starts from the point $(0,8)$ metre and moves with uniform velocity of $\vec{v}=3 \hat{i} \,m / s$. What is the angular momentum of the particle after $5 \,s$ about origin is ........... $kg m ^2 / s$ (mass of particle is $1 \,kg$ )?

A particle is moving along a straight line with increasing speed. Its angular momentum about a fixed point on this line ............

Obtain $\tau = I\alpha $ from angular momentum of rigid body.